Suppose Y = X1 + X2, where Xi ? exp(1) for i = 1, 2. Determine the distribution of Y
This is a probability question. Please be thorough and
detailed.
3. (8 pts.) Suppose that Xi ~ Exp(A) and X2 ~ Exp(A2) where λ1 and λ2 are positive con- λ2, but do assume that Xi and X2 are independent. Compute stants. Do not assume λι P(X1 < X2). Now note that the probability you just computed is in fact P(Xmin(XI, X2)). This suggests the following generalization. Suppose we have a collection of N independent ex- ponential random variables, X1, X2,...
Suppose X1, X2, Xz~exp(1) and they are independent. (a) Compute the cdf of X1 (b) Let Y- max(Xi, X2, X3). Find the cdf of Y (c) Derive the pdf of Y
7. Let X1 and X2 be two iid exp(A) random variables. Set Yi Xi - X2 and Y2 X + X2. Determine the joint pdf of Y and Y2, identify the marginal distributions of Yi and Y2, and decide whether or not Yi and Y2 are independent [10)
Suppose that (X1, X2) N (0,0,1,1,0). It follows from this that the joint PDF of (X1, X2) is given by Ixvx:(21,2) = exp (1} (27 +23)) Furthermore, if 1 Y Tā (X1 + X2) and Y2 (X1 - X2) Then (Y1,Y) ~ N(0,0,1,1,0) as well. (a) If X1 <X2, what are the possible values of Y¡ and Y2? (b) If Y, <0, what are the possible values of Xi and X,? (c) What is the marginal distribution of Yg? (d)...
1. Let Xi, X2, X, be a 1.1.d. sample form Exp(1), and Y = Σ=i Xi. (a) Use CLT to get a large sample distribution of Y (b) For n = 100, give an approximation for P(Y > 100) (c) Let X be the sample mean, then approximate P(1.1 < X < 1.2) for n = 100.
Let λ >0 and suppose that X1,X2,...,Xn be i.i.d. random variables with Xi∼Exp(λ). Find the PDF of X1+···+Xn. Use convolution formula and prove by induction
4. Suppose Xi, X2, X3 ~exp(1) and they are independent (a) Compute the edf of X (b) Let Y max(Xi, X2, X3). Find the cdf of Y. (c) Derive the pdf of Y
2. Let Xi exp(1) and X2 ~ variables with rate 1. Let: erp(1) be independent and identically-distributed exponential random (a) What is the cdf of X1? b) What is the joint pdf of (Xi, X2)? (c) What is the joint pdf of (Y, Z)? d) What is the marginal pdf of z?
Let Xi and X2 independent random variables, with distribution functions F1, and F2, respectively Let Y a Bernoulli random variable with parameter p. Suppose that Y, X1 and X2 are independent. Proof using the de finition of distribution function that the the distribution function of Z =Y Xit(1-Y)X2 is F = pF14(1-p)F2 Don't use generatinq moment functions, characteristic functions) Xi and X2 independent random variables, with distribution functions F1, and F2, respectively Let Y a Bernoulli random variable with parameter...
3. (25 pts.) Let X1, X2, X3 be independent random variables such that Xi~ Poisson (A), i 1,2,3. Let N = X1 + X2+X3. (a) What is the distribution of N? (b) Find the conditional distribution of (X1, X2, X3) | N. (c) Now let N, X1, X2, X3, be random variables such that N~ Poisson(A), (X1, X2, X3) | N Trinomial(N; pi,p2.ps) where pi+p2+p3 = 1. Find the unconditional distribution of (X1, X2, X3).
3. (25 pts.) Let X1,...